vibrations and waves

what is the general equation of motion

ẍ = -w² x

what does w² equal

k/m

what is the x(t) general equation when initial velocity = 0

A sin(wt + φ)

what is x(t) general equation if initial velocity is non zero

A cos(wt + φ)

what is the energy of a SHM oscillator

½ K A²

what is damoed harmonic motion

has damping forces which oppose motion and dissipate energy

what is damping force

F = -bv

what is the equation of damped harmonic motion

ẍ + γẋ + w0² x = 0

what is γ

b/m [s^-1]

what is w0²

k/m [w0 = rad s^-1]

what is the equation for light damping

e^(-γt/2) A cos(wt + φ)

what is the w0 for lamp damping

w0² > γ²/4

what is the equation of motion heavy damping

e^(-γt/2) [ Ae^α + Be^-α]

α = sqrt ( γ²/4 - w0² )

what is the w0 for heavy damping

γ²/4 > w0²

what is the equation of motion for critical damping

e^(-γt/2) [ A + Bt]

what is the w0 for critical damping

γ²/4 = w0²

what is the energy of light damping

½ k A² e^(-γt)

what is the quality factor

Q = w0 / γ

time of relaxation / period of oscillation

how to find w from Q

w = w0 sqrt( 1- 1/(4Q²) )

what is the fractional energy change per cycle

E1 - E2 /E1 = 2π/Q

what is the general solution of forced damp harmonic motion

e^(-γt/2) A cos(wt + φ)

equation of forced damped harmonic motion

steady state amplitude of force oxsillation graph